Chapman function
[file:Chapman function.svg|thumb|300px|right|Graph of ch(x, z)]
A Chapman function, denoted, describes the integration of an atmospheric parameter along a slant path on a spherical Earth, relative to the vertical or zenithal case. It applies to any physical quantity with a concentration decreasing exponentially with increasing altitude. At small angles, the Chapman function is approximately equal to the secant function of the zenith angle,.
The Chapman function is named after Sydney Chapman, who introduced the function in 1931.
It has been applied for absorption and the ionosphere.
Definition
In an isothermal model of the atmosphere, the density varies exponentially with altitude according to the Barometric formula:where denotes the density at sea level and the so-called scale height.
The total amount of matter traversed by a vertical ray starting at altitude towards infinity is given by the integrated density
For inclined rays having a zenith angle, the integration is not straight-forward due to the non-linear relationship between altitude and path length when considering the
curvature of Earth. Here, the integral reads
where we defined .
The Chapman function is defined as the ratio between slant depth and vertical column depth. Defining, it can be written as
Representations
A number of different integral representations have been developed in the literature. Chapman's original representation readsHuestis developed the representation
which does not suffer from numerical singularities present in Chapman's representation.
Special cases
For, the Chapman function reduces toHere, refers to the modified Bessel function of the second kind of the first order. For large values of, this can further be approximated by
For and, the Chapman function converges to the secant function:
In practical applications related to the terrestrial atmosphere, where, is a good approximation for zenith angles up to 60° to 70°, depending on the accuracy required.
Approximations
For and, the approximationis accurate to 2 % at and to 0.1 % at. The accuracy improves with increasing.